The following questions are about the function f(x,y)=x^{2}e^{2xy}.

Step 1
If f(x,y) is a function of two variables x and y, then
${\displaystyle f_x(x,y)=\frac{\partial f}{\partial x}}$
${\displaystyle f_y(x,y)=\frac{\partial f}{\partial y}}$
The equation of the tangent plane of function f at the point ${\displaystyle (x_0,y_0)}$ is
${\displaystyle z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)}$
Step 2
The value of the given function at (2,0) is
${\displaystyle f(2,0)={\left(2\right)}^2{e}^{2\left(2\right)\left(0\right)}}$
${\displaystyle =4e^0}$
=4(1)
=4
The equation of the tangent plane of function f at the point (2,0) is
${\displaystyle z=f(2,0)+f_x(2,0)(x-2)+f_y(2,0)(y-0)}$
z=4+4(x-2)+16(y)
z=4+4x-8+16y
z=4x+16y-4